Calculus-Based Problems

The displacement of an object is determined by the following function:

$$ s=2 t^{3}-9 t^{2}+12 t+4 $$

where $s$ is the displacement in metres, and $t$ the time elapsed in seconds. Determine

(i) the times when the object comes to rest,

(ii) the time when the acceleration is zero,

(iii) the object's velocity when its acceleration is zero,

(iv) the object's accelerations when its velocity is zero. [4]

A car accelerates from a standing start. If the mass of the car is $m$, and the car is driven at constant driving power $P$, find an expression for the velocity of the car $v$ as a function of distance travelled from a standing start, $x$. Ignore resistive effects and inefficiencies in power transmission. [4]

A cup of tea cools from $30.2^{\circ} \mathrm{C}$ to $29.7^{\circ} \mathrm{C}$ in 1 minute, in an ambient temperature of $20.0^{\circ} \mathrm{C}$. Assuming the tea cools at a rate directly proportional to the temperature difference between the tea and the surroundings, calculate how long it will take for the tea to cool from $24.0^{\circ} \mathrm{C}$ to $23.0^{\circ} \mathrm{C}$. [5]

; Emmy walks into a lift with a set of (bathroom) scales. She stands on the scales, presses the button for the $30^{\text {th }}$ floor, and starts a timer as the lift begins to move. She notices that the reading on the scales varies with time, according to the equation below:

$$ m(t)=60\left(1+\frac{t}{10}-\frac{t^{2}}{100}\right) $$

(i) Write down an expression for the acceleration of the lift as a function of time.

(ii) How fast is the lift moving after 10 seconds?

(iii) After the initial 10 seconds, the lift decelerates at a constant rate until it arrives at the $30^{\text {th }}$ floor. Given that the $30^{\text {th }}$ floor is $100 \mathrm{~m}$ above the ground, calculate the minimum value of the mass reading (in $\mathrm{kg}$) shown on the scales during this deceleration. [4]

The stiffness, $S$, of a beam of rectangular cross-section, with width $w$, and thickness $t$, is directly proportional to its width and the cube of its thickness, $\left(t^{3}\right)$; that is, stiffness, $S \propto w t^{3}$. Determine the cross-sectional dimensions of the stiffest rectangular cross section wooden beam that can be cut from a log of diameter $20 \mathrm{~cm}$. [3]

An aeroplane flies over an observer at speed $v$ and at a fixed height of $3000 \mathrm{~m}$. After some time, the observer sees the plane at an angle of $60^{\circ}$ above the horizontal, which is decreasing at a rate of $0.09 \mathrm{rad} \mathrm{s}^{-1}$. Calculate

(i) the distance from the observer to the plane

(ii) the speed of the plane

(iii) the speed at which the plane is receding along the observer's line of sight. [5]

Before a device is fully charged, the current flowing into it at a time $t < t_{\text{fully charged}}$ is given by $$I = I_0 - \alpha t$$ where $I_0$ is the initial current.

1. (i) How long does it take to charge it, and
2. (ii) how much charge is accumulated?